{"id":"openai-math/Quasipolynomial-Bounds-for-Arithmetic-Progressions","pdf":"https://github.com/openai/math/blob/main/preprints/Quasipolynomial-Bounds-for-Arithmetic-Progressions-September-23-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/Quasipolynomial-Bounds-for-Arithmetic-Progressions-September-23-2026","tags":["additive-combinatorics","arithmetic-progressions","erdos-problems","combinatorics"],"score":0,"title":"Quasipolynomial Bounds for Arithmetic Progressions","domain":"Combinatorics","family":"159","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.CO","summary":"We prove Erdős's conjecture that every set of positive integers with divergent reciprocal sum contains arithmetic progressions of every finite length. More quantitatively, for every fixed k ≥ 3, we show $`\\displaystyle r_k(N)\\le C_kN\\exp\\bigl(-c_k(\\log N)^{\\varepsilon_k}\\bigr)`$ with $`C_k,c_k,\\varepsilon_k\\gt 0`$, where $`r_k(N)`$ is the largest size of a subset of $`\\{1,\\ldots,N\\}`$ with no nonconstant k -term arithmetic progression.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":3,"published":"2026-09-23","reactions":{},"discussion":[{"url":"https://gilkalai.wordpress.com/2026/10/07/updates-sharing-ai-progress-on-mathematics-amazing-and-my-lecture-plans/#:~:text=any%20set%20of%20positive%20integers%20with%20divergent%20reciprocal%20sum%20contains%20arbitrarily%20long%20arithmetic%20progressions","date":"2026-10-07","kind":"web","site":"Combinatorics and more","text":"any set of positive integers with divergent reciprocal sum contains arbitrarily long arithmetic progressions, with quasipolynomial quantitative bounds in Szemerédi’s theorem. Sidorenko’s conjecture. I described Sidorenko’s conjecture , which predicts that a fixed bipartite graph has at least the…","added":"2026-10-24","title":"Updates: Sharing AI progress on mathematics (amazing!); and my lecture plans"},{"url":"https://x.com/AndrewLott1331/status/2108189504259727838","date":"2026-10-08","kind":"x","added":"2026-10-24","author":"Andrew Lott","handle":"AndrewLott1331"},{"url":"https://x.com/thomasfbloom/status/2108191863572185473","date":"2026-10-08","kind":"x","added":"2026-10-24","author":"Thomas Bloom","handle":"thomasfbloom"}],"formalized":false,"family_title":"Erdős’s reciprocal-sum conjecture and quasipolynomial Szemerédi bounds"}